39 research outputs found
An iterative-bijective approach to generalizations of Schur's theorem
We start with a bijective proof of Schur's theorem due to Alladi and Gordon
and describe how a particular iteration of it leads to some very general
theorems on colored partitions. These theorems imply a number of important
results, including Schur's theorem, Bressoud's generalization of a theorem of
G\"ollnitz, two of Andrews' generalizations of Schur's theorem, and the
Andrews-Olsson identities.Comment: 16 page
Double series representations for Schur's partition function and related identities
We prove new double summation hypergeometric -series representations for
several families of partitions, including those that appear in the famous
product identities of G\"ollnitz, Gordon, and Schur. We give several different
proofs for our results, using bijective partitions mappings and modular
diagrams, the theory of -difference equations and recurrences, and the
theories of summation and transformation for -series. We also consider a
general family of similar double series and highlight a number of other
interesting special cases.Comment: 19 page
A unifying combinatorial approach to refined little G\"ollnitz and Capparelli's companion identities
Berkovich-Uncu have recently proved a companion of the well-known
Capparelli's identities as well as refinements of Savage-Sills' new little
G\"ollnitz identities. Noticing the connection between their results and
Boulet's earlier four-parameter partition generating functions, we discover a
new class of partitions, called -strict partitions, to generalize their
results. By applying both horizontal and vertical dissections of Ferrers'
diagrams with appropriate labellings, we provide a unified combinatorial
treatment of their results and shed more lights on the intriguing conditions of
their companion to Capparelli's identities.Comment: This is the second revision submitted to JCTA in June, comments are
welcom
A new four parameter q-series identity and its partition implications
We prove a new four parameter q-hypergeometric series identity from which the
three parameter key identity for the Goellnitz theorem due to Alladi, Andrews,
and Gordon, follows as a special case by setting one of the parameters equal to
0. The new identity is equivalent to a four parameter partition theorem which
extends the deep theorem of Goellnitz and thereby settles a problem raised by
Andrews thirty years ago. Some consequences including a quadruple product
extension of Jacobi's triple product identity, and prospects of future research
are briefly discussed.Comment: 25 pages, in Sec. 3 Table 1 is added, discussion is added at the end
of Sec. 5, minor stylistic changes, typos eliminated. To appear in
Inventiones Mathematica